Simplify (4x^3-3 square root of 3x)/(3 square root of 3x^2)

$\frac{4 x^{3} - 3 \sqrt{3 x}}{3 \sqrt{3 x^{2}}}$

Step 1

Simplify the numerator.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x}$ as ${(3 x)}^{\frac{1}{2}}$ .

$\frac{4 x^{3} - 3 {(3 x)}^{\frac{1}{2}}}{3 \sqrt{3 x^{2}}}$

Apply the product rule to $3 x$ .

$\frac{4 x^{3} - 3 (3^{\frac{1}{2}} x^{\frac{1}{2}})}{3 \sqrt{3 x^{2}}}$

Multiply $- 3 \cdot 3^{\frac{1}{2}}$ .

Factor out negative.

$\frac{4 x^{3} - (3 \cdot 3^{\frac{1}{2}}) x^{\frac{1}{2}}}{3 \sqrt{3 x^{2}}}$

Raise $3$ to the power of $1$ .

$\frac{4 x^{3} - (3^{1} \cdot 3^{\frac{1}{2}}) x^{\frac{1}{2}}}{3 \sqrt{3 x^{2}}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{4 x^{3} - 3^{1 + \frac{1}{2}} x^{\frac{1}{2}}}{3 \sqrt{3 x^{2}}}$

Write $1$ as a fraction with a common denominator.

$\frac{4 x^{3} - 3^{\frac{2}{2} + \frac{1}{2}} x^{\frac{1}{2}}}{3 \sqrt{3 x^{2}}}$

Combine the numerators over the common denominator.

$\frac{4 x^{3} - 3^{\frac{2 + 1}{2}} x^{\frac{1}{2}}}{3 \sqrt{3 x^{2}}}$

Add $2$ and $1$ .

$\frac{4 x^{3} - 3^{\frac{3}{2}} x^{\frac{1}{2}}}{3 \sqrt{3 x^{2}}}$

Factor $x^{\frac{1}{2}}$ out of $4 x^{3} - 3^{\frac{3}{2}} x^{\frac{1}{2}}$ .

Factor $x^{\frac{1}{2}}$ out of $4 x^{3}$ .

$\frac{x^{\frac{1}{2}} (4 x^{\frac{5}{2}}) - 3^{\frac{3}{2}} x^{\frac{1}{2}}}{3 \sqrt{3 x^{2}}}$

Factor $x^{\frac{1}{2}}$ out of $- 3^{\frac{3}{2}} x^{\frac{1}{2}}$ .

$\frac{x^{\frac{1}{2}} (4 x^{\frac{5}{2}}) + x^{\frac{1}{2}} (- 3^{\frac{3}{2}})}{3 \sqrt{3 x^{2}}}$

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} (4 x^{\frac{5}{2}}) + x^{\frac{1}{2}} (- 3^{\frac{3}{2}})$ .

$\frac{x^{\frac{1}{2}} (4 x^{\frac{5}{2}} - 3^{\frac{3}{2}})}{3 \sqrt{3 x^{2}}}$

Step 2

Simplify the denominator.

Reorder $3$ and $x^{2}$ .

$\frac{x^{\frac{1}{2}} (4 x^{\frac{5}{2}} - 3^{\frac{3}{2}})}{3 \sqrt{x^{2} \cdot 3}}$

Pull terms out from under the radical.

$\frac{x^{\frac{1}{2}} (4 x^{\frac{5}{2}} - 3^{\frac{3}{2}})}{3 x \sqrt{3}}$

Step 3

Move $x^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{4 x^{\frac{5}{2}} - 3^{\frac{3}{2}}}{3 x \sqrt{3} x^{- \frac{1}{2}}}$

Step 4

Multiply $x$ by $x^{- \frac{1}{2}}$ by adding the exponents.

Move $x^{- \frac{1}{2}}$ .

$\frac{4 x^{\frac{5}{2}} - 3^{\frac{3}{2}}}{3 (x^{- \frac{1}{2}} x) \sqrt{3}}$

Multiply $x^{- \frac{1}{2}}$ by $x$ .

Raise $x$ to the power of $1$ .

$\frac{4 x^{\frac{5}{2}} - 3^{\frac{3}{2}}}{3 (x^{- \frac{1}{2}} x^{1}) \sqrt{3}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{4 x^{\frac{5}{2}} - 3^{\frac{3}{2}}}{3 x^{- \frac{1}{2} + 1} \sqrt{3}}$

Write $1$ as a fraction with a common denominator.

$\frac{4 x^{\frac{5}{2}} - 3^{\frac{3}{2}}}{3 x^{- \frac{1}{2} + \frac{2}{2}} \sqrt{3}}$

Combine the numerators over the common denominator.

$\frac{4 x^{\frac{5}{2}} - 3^{\frac{3}{2}}}{3 x^{\frac{- 1 + 2}{2}} \sqrt{3}}$

Add $- 1$ and $2$ .

$\frac{4 x^{\frac{5}{2}} - 3^{\frac{3}{2}}}{3 x^{\frac{1}{2}} \sqrt{3}}$

Step 5

Multiply $\frac{4 x^{\frac{5}{2}} - 3^{\frac{3}{2}}}{3 x^{\frac{1}{2}} \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{4 x^{\frac{5}{2}} - 3^{\frac{3}{2}}}{3 x^{\frac{1}{2}} \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 6

Combine and simplify the denominator.

Multiply $\frac{4 x^{\frac{5}{2}} - 3^{\frac{3}{2}}}{3 x^{\frac{1}{2}} \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{(4 x^{\frac{5}{2}} - 3^{\frac{3}{2}}) \sqrt{3}}{3 x^{\frac{1}{2}} \sqrt{3} \sqrt{3}}$

Move $\sqrt{3}$ .

$\frac{(4 x^{\frac{5}{2}} - 3^{\frac{3}{2}}) \sqrt{3}}{3 x^{\frac{1}{2}} (\sqrt{3} \sqrt{3})}$

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{(4 x^{\frac{5}{2}} - 3^{\frac{3}{2}}) \sqrt{3}}{3 x^{\frac{1}{2}} ({\sqrt{3}}^{1} \sqrt{3})}$

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{(4 x^{\frac{5}{2}} - 3^{\frac{3}{2}}) \sqrt{3}}{3 x^{\frac{1}{2}} ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{(4 x^{\frac{5}{2}} - 3^{\frac{3}{2}}) \sqrt{3}}{3 x^{\frac{1}{2}} {\sqrt{3}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{(4 x^{\frac{5}{2}} - 3^{\frac{3}{2}}) \sqrt{3}}{3 x^{\frac{1}{2}} {\sqrt{3}}^{2}}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{(4 x^{\frac{5}{2}} - 3^{\frac{3}{2}}) \sqrt{3}}{3 x^{\frac{1}{2}} {(3^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{(4 x^{\frac{5}{2}} - 3^{\frac{3}{2}}) \sqrt{3}}{3 x^{\frac{1}{2}} \cdot 3^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$\frac{(4 x^{\frac{5}{2}} - 3^{\frac{3}{2}}) \sqrt{3}}{3 x^{\frac{1}{2}} \cdot 3^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{(4 x^{\frac{5}{2}} - 3^{\frac{3}{2}}) \sqrt{3}}{3 x^{\frac{1}{2}} \cdot 3^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{(4 x^{\frac{5}{2}} - 3^{\frac{3}{2}}) \sqrt{3}}{3 x^{\frac{1}{2}} \cdot 3^{1}}$

Evaluate the exponent.

$\frac{(4 x^{\frac{5}{2}} - 3^{\frac{3}{2}}) \sqrt{3}}{3 x^{\frac{1}{2}} \cdot 3}$

Step 7

Simplify the expression.

Multiply $3$ by $3$ .

$\frac{(4 x^{\frac{5}{2}} - 3^{\frac{3}{2}}) \sqrt{3}}{9 x^{\frac{1}{2}}}$

Reorder factors in $\frac{(4 x^{\frac{5}{2}} - 3^{\frac{3}{2}}) \sqrt{3}}{9 x^{\frac{1}{2}}}$ .

$\frac{\sqrt{3} (4 x^{\frac{5}{2}} - 3^{\frac{3}{2}})}{9 x^{\frac{1}{2}}}$

Simplify (4x^3-3 square root of 3x)/(3 square root of 3x^2)

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